Let Y;, i = 1,...,n be independent N(Bo + YiB1, o²) and let W1,..., Wm be independent N(Y + w;Y1, o²), with known, and non-random, numbers y1,..., Yn, Wi,..., Wn. Construct a test of Ho : B1 = Y1 against H.: B1 7 Y1.
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- Let x and y be in Z, not both zero, then x2+y2Z+.Let X1, . . . , Xm i.i.d.∼ N (μ1, 1) and Y1, . . . , Yn i.i.d.∼ N (μ2, 1).Suppose that these two samples are independent. We would like to testH0 : μ1 = μ2, H1 : μ1 != μ2 using the likelihood ratio test.An instructor has given a short quiz consisting of two parts. For a randomly selected student, let X = the number of points earned on the first part and Y = the number of points earned on the second part. Suppose that the joint pmf of X and Y is given in the accompanying table. y p(x, y) 0 5 10 15 x 0 0.01 0.06 0.02 0.10 5 0.04 0.14 0.20 0.10 10 0.01 0.15 0.16 0.01 (a) If the score recorded in the grade book is the total number of points earned on the two parts, what is the expected recorded score E(X + Y)? (Enter your answer to one decimal place.)(b) If the maximum of the two scores is recorded, what is the expected recorded score? (Enter your answer to two decimal places.)
- Let X1 and X2 be independent random variables for which P(Xi = 1) = 2/5 and P(Xi = 2) = 3/5 . Define U = X1 + X2 and V = X1 x X2. Calculate Cor(U, V )In a random sample of n pairs of values of X and Y,(xi, yj) occurs fij times for i = 1, 2, ... ,r and j = 1, 2, ... , c.Letting fi· denote the number of pairs where X takes onthe value xi and f·j the number of pairs where Y takeson the value yj, write a formula for the coefficient ofcorrelation.An instructor has given a short quiz consisting of two parts. For a randomly selected student, let X = the number of points earned on the first part and Y = the number of points earned on the second part. Suppose that the joint pmf of X and Y is given in the accompanying table. y p(x, y) 0 5 10 15 x 0.01 0.06 0.02 0.10 5 0.04 0.16 0.20 0.10 10 0.01 0.15 0.14 0.01 (a) Compute the covariance for X and Y. (Round your answer to two decimal places.)Cov(X, Y) = (b) Compute ? for X and Y. (Round your answer to two decimal places.)? =
- Let X1, . . . , Xn i.i.d. U([θ1, θ2]), i.e., X1, . . . , Xn are independent and follow a uniform distribution on the interval [θ1, θ2] for θ1, θ2 ∈ R and θ1 < θ2. Find an estimator for θ1 and θ2 using the method of moments.Let Yi , i = 1, . . . , n be independent N (β0 + yiβ1, σ^2 ) and let W1, . . . , Wm be independent N (γ0 + wiγ1, σ^2 ), with known, and non-random, numbers y1, . . . , yn, w1, . . . , wn. Construct a test of H0 : β1 = γ1 against Ha : β1 /= γ1.If X1, X2, ... , Xn constitute a random sample of size nfrom a geometric population, show that Y = X1 + X2 +···+ Xn is a sufficient estimator of the parameter θ.
- An instructor has given a short quiz consisting of two parts. For a randomly selected student, let X = the number of points earned on the first part and Y = the number of points earned on the second part. Suppose that the joint pmf of X and Y is given in the accompanying table. y p(x, y) 0 5 10 15 x 0 0.01 0.06 0.02 0.10 5 0.04 0.16 0.20 0.10 10 0.01 0.15 0.14 0.01 (a) Compute the covariance for X and Y. (Round your answer to two decimal places.)Cov(X, Y) = (b) Compute ρ for X and Y. (Round your answer to two decimal places.)ρ =Let X, Y, and Z be jointly distributed random variables. Prove that Cov(X + Y, Z) = Cov(X, Z) + Cov(Y, Z).Q. For any random variables X and Y and the constant a, b, c and d show that Cov(aX + b, cY + d) = ac Cov(X, Y).